Average Error: 6.1 → 0.7
Time: 12.9s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -4.70661221182111 \cdot 10^{+192}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t - z}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 4.197629598677036 \cdot 10^{+144}:\\ \;\;\;\;x + \frac{y \cdot z - y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \leq -4.70661221182111 \cdot 10^{+192}:\\
\;\;\;\;x - \frac{y}{\frac{a}{t - z}}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \leq 4.197629598677036 \cdot 10^{+144}:\\
\;\;\;\;x + \frac{y \cdot z - y \cdot t}{a}\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{z - t}{a}\\

\end{array}
(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* y (- z t)) -4.70661221182111e+192)
   (- x (/ y (/ a (- t z))))
   (if (<= (* y (- z t)) 4.197629598677036e+144)
     (+ x (/ (- (* y z) (* y t)) a))
     (+ x (* y (/ (- z t) a))))))
double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / a);
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * (z - t)) <= -4.70661221182111e+192) {
		tmp = x - (y / (a / (t - z)));
	} else if ((y * (z - t)) <= 4.197629598677036e+144) {
		tmp = x + (((y * z) - (y * t)) / a);
	} else {
		tmp = x + (y * ((z - t) / a));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.1
Target0.7
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 y (-.f64 z t)) < -4.70661221182111e192

    1. Initial program 27.1

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary64_829727.5

      \[\leadsto x + \frac{y \cdot \left(z - t\right)}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

    4. Applied associate-/r*_binary64_820627.5

      \[\leadsto x + \color{blue}{\frac{\frac{y \cdot \left(z - t\right)}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}}\]

    5. Simplified7.2

      \[\leadsto x + \frac{\color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}}{\sqrt[3]{a}}\]
    6. Using strategy rm

    7. Applied clear-num_binary64_82617.2

      \[\leadsto x + \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}}}\]

    8. Simplified0.5

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z - t}}}\]

    9. Taylor expanded around -inf 27.1

      \[\leadsto x + \color{blue}{-1 \cdot \frac{t \cdot y - z \cdot y}{a}}\]

    10. Simplified1.1

      \[\leadsto x + \color{blue}{\frac{-y}{\frac{a}{t - z}}}\]

    if -4.70661221182111e192 < (*.f64 y (-.f64 z t)) < 4.1976295986770361e144

    1. Initial program 0.4

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied sub-neg_binary64_82550.4

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(z + \left(-t\right)\right)}}{a}\]

    4. Applied distribute-rgt-in_binary64_82120.4

      \[\leadsto x + \frac{\color{blue}{z \cdot y + \left(-t\right) \cdot y}}{a}\]

    5. Simplified0.4

      \[\leadsto x + \frac{z \cdot y + \color{blue}{\left(-t \cdot y\right)}}{a}\]

    if 4.1976295986770361e144 < (*.f64 y (-.f64 z t))

    1. Initial program 19.4

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity_binary64_826219.4

      \[\leadsto x + \frac{y \cdot \left(z - t\right)}{\color{blue}{1 \cdot a}}\]

    4. Applied times-frac_binary64_82681.9

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{z - t}{a}}\]

    5. Simplified1.9

      \[\leadsto x + \color{blue}{y} \cdot \frac{z - t}{a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -4.70661221182111 \cdot 10^{+192}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t - z}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 4.197629598677036 \cdot 10^{+144}:\\ \;\;\;\;x + \frac{y \cdot z - y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))